Branching Random Walks Conditioned on Particle Numbers

Branching Random Walks on Graphs

We study a new distributed randomized information propagation mechanism in networks that we call a branching random walk (BRW). BRW is a generalization of the well-studied “standard” random walk which is a fundamental primitive useful in a wide variety of network applications ranging from token management and load balancing to search, routing, information propagation and gossip. BRW is paramete...

Branching Random Walks

We study a generalized branching random walk where particles breed at a rate which depends on the number of neighbouring particles. Under general assumptions on the breeding rates we prove the existence of a phase where the population survives without exploding. We construct a non trivial invariant measure for this case.

Anisotropic Branching Random Walks on Homogeneous Trees

Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundary of the tree. The random subset 3 of the boundary consisti...

Minima in Branching Random Walks

Given a branching random walk, let Mn be the minimum position of any member of the n’th generation. We calculate EMn to within O(1) and prove exponential tail bounds for P {|Mn − EMn| > x}, under quite general conditions on the branching random walk. In particular, together with work of [8], our results fully characterize the possible behavior of EMn when the branching random walk has bounded b...

Branching Processes and Random Walks